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4.5: The Biconditional

We introduce one more connective into sentence logic. Often we will want to study cases which involve a conjunction of the form (X⊃Y)&(Y⊃X). This truth function of X and Y occurs so often in logic that we give it its own name, the Biconditional, which we write as X≡Y. Working out the truth table of (X⊃Y)&(Y⊃X) we get as our definition of the biconditional:

Truth table Definition of ≡

X Y X≡Y
t t t
t f f
f t f
f f t

   Because a biconditional has a symmetric definition, we don't have different names for its components. We just call them 'components'. You will remember this definition most easily by remembering that a biconditional is true if both components have the same truth value (both true or both false), and it is false if the two components have different truth values (one true, the other false). We read the biconditional X≡Y with the words 'X if and only if Y. With the biconditional, we get into much less trouble with transcriptions between English and sentence logic than we did with the conditional.
   Given the way we define '≡', we have the logical equivalence:

The Law of the Biconditiuml (B): X≡Y is logically equivalent to (X⊃Y)&(Y⊃X).

   Remember that the conditional, X⊃Y, is a logical truth just in case the corresponding argument, "X. Therefore Y, is valid. Likewise, there is something interesting we can say about the biconditional, X≡Y, being a logical truth:

X≡Y is a logical truth if and only if X and Y are logically equivalent.

Can you see why this is true? Suppose X≡Y is a logical truth. This means that in every possible case (for every assignment of truth values to sentence letters) X≡Y is true. But X≡Y is true only when its two components have the same truth value. So in every possible case, X and Y have the same truth value, which is just what we mean by saying that they are logically equivalent. On the other hand, suppose that X and Y are logically equivalent. This just means that in every possible case they have the same truth value. But when X and Y have the same truth value, X≡Y is true. So in every possible case X≡Y is true, which is just what is meant by saying that X≡Y is a logical truth.

Exercise \(\PageIndex{1}\)

4-4. In section 1-6 1 gave rules of formation and valuation for sentence logic. Now that we have extended sentence logic to include the connectives '3' and '=', these rules also need to be extended. Write the full rules of formation and valuation for sentence logic, where sentence logic may now use all of the connectives '-', '&', 'v', '', and ''. In your rules, also provide for three and more place conjunctions and disjunctions as described in section 3-2 in the discussion of the associative law. 

4-5. Follow the same instructions as for exercise 4-2.

a) A⊃B         b) A⊃~B     c)   A≡B         d) A≡~B
    B                     B             Av~B            Av~B
    A                    ~A             B                  AvB

e) (AvB)⊃(A&C)         f) (AvB)≡(Av-C)
          CvA                         ~BvC         
           ~C                           AvC

4-6. For each of the following sentences, establish whether it is a logical truth, a contradiction, or neither. Use the laws of logical equivalence in chapter 3 and sections 4-3 and 4-4, and use the fact that a biconditional is a logical truth if and only if its components are logically equivalent.

4-7. Discuss how you would transcribe 'unless' into sentence logic. Experiment with some examples, trying out the use of 'v', '>', and I='. Bear in mind that one connective might work well for one example, another connective for another example. As you work, pay attention to whether or not the compound English sentences you choose as examples are truth functional. Report the results of your research by giving the following:

a) Give an example of a compound English sentence using 'unless' which seems to be nontruth functional, explaining why it is not truth functional.

b) Give an example of a compound English sentence using 'unless' which seems to be truth functional, explaining why it is truth functional.

c) Give one example each of English sentences using 'unless' which can be fairly well transcribed into sentence logic using 'v', '', '', ' giving the transcriptions into sentence logic.

4-8. Transcribe the following sentences into sentence logic, using the given transcription guide:

A: Adam loves Eve.                 D: Eve has dark eyes.
B: Adam is blond.                   E: Eve loves Adam.
C:Eve is clever.

a) If Eve has dark eyes, then Adam does not love her.

b) Adam loves Eve if she has dark eyes.

c) If Adam loves Eve, Eve does not love Adam.

d) Eve loves Adam only if he is not blond.

e) Adam loves Eve if and only if she has dark eyes.

f) Eve loves Adam provided he is blond.

g) Provided she is clever, Adam loves Eve.

h) Adam does not love Eve unless he is blond.

i) Unless Eve is clever, she does not love Adam.

j) If Adam is blond, then he loves Eve only if she has dark eyes.

k) If Adam is not blond, then he loves Eve whether or not she has dark eyes.

l) Adam is blond and in love with Eve if and only if she is clever.

m) Only if Adam is blond is Eve both clever and in love with Adam.

4-9. Consider the following four different kinds of nontruth functional connectives that can occur in English:

a) Connectives indicating connections (causal, intentional, or conventional)

b) Modalities (what must, can, or is likely to happen)

c) So-called "propositional attitudes," having to do with what people know, believe, think, hope, want, and the like

d) Temporal connectives, having to do with what happens earlier, later, or at the same time as something else.

Give as many English connectives as you can in each category. Keep in mind that some connectives will go in more than one category. ('Since' is such a connective. What two categories does it go into?) To get you started, here are some of these connectives: 'because', 'after', 'more likely than', 'Adam knows that', 'Eve hopes that'.

Chapter summary Exercise

Give brief explanations for each of the following. As usual, check your explanations against the text to make sure you get them right, and keep them in your notebook for reference and review.

a) Valid

b) Invalid

c) Counterexample

d) Sound

e) Conditional

f) Biconditional

g) Law of the Conditional

h) Law of Contraposition

i) Law of the Biconditional

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